During the passage of a proton bunch through an electron cloud, a complicated electron density modulation arises, with characteristic ring or stripe patterns that move radially outward along the bunch. We present simulation results for field-free and dipole regions, which reveal the morphology and main features of this phenomenon, explain the physical origin of the stripes in either case, and discuss the dependence on key parameters

Franchetti, G

Zimmermann, F

Franchetti, Giuliano

Zimmermann, F.

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Electron-cloud Intrabunch Density Modulation

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LHC-PROJECT-REPORT-110127Aug2008F. Zimmermann, G. Franchetti1)EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHEuropean Laboratory for Particle PhysicsELECTRON-CLOUD INTRABUNCH DENSITY MODULATIONDuring the passage of a proton bunch through an electron cloud, a complicated electron density modulationarises, with characteristic ring or stripe patterns that move radially outward along the bunch. We presentsimulation results for field-free and dipole regions, which reveal the morphology and main features of thisphenomenon, explain the physical origin of the stripes in either case, and discuss the dependence on keyparameters.Large Hadron Collider ProjectAbstractLHC Project Report 1101CERN,CH-1211 Geneva 23Switzerland1) GSI, Darmstadt, GermanyCERN, Geneva, SwitzerlandPresented atEPAC'08, 11th European Particle Accelerator Conference, Genoa, Italy - June 23-27, 2008Electron-Cloud Intrabunch Density ModulationF. Zimmermann, CERN, Geneva, Switzerland; G. Franchetti, GSI, Darmstadt, GermanyAbstractDuring the passage of a proton bunch through an electroncloud, a complicated electron density modulation arises,with characteristic ring or stripe patterns that move radiallyoutward along the bunch. We present simulation results forfield-free and dipole regions, which reveal the morphologyand main features of this phenomenon, explain the physicalorigin of the stripes in either case, and discuss the depen-dence on key parameters.INTRODUCTIONWhen a proton or positron bunch passes through anelectron cloud generated by preceding bunches, the cloudelectrons are attracted towards the transverse center of thebunch or “pinched”, resulting in regions of high electrondensity inside the bunch. This electron-cloud “pinch” givesrise to an incoherent betatron tune shift which varies withthe longitudinal position and with transverse amplitude.Combined with synchrotron motion and together with thenon-uniform distribution of the electron cloud around astorage ring (in the SPS, for example, the electron cloudbuilds up preferentially inside the dipole magnets [1]), thistune shift in turn leads to the excitation of betatron andsynchro-betatron resonances [2, 3], as well as to “scatter-ing” off these resonances [3]. For the LHC proton beam inthe PS, SPS and LHC itself, these effects can be significant[4]. Some of their characteristics resemble space-chargephenomena [3, 5, 6].Early models of the electron pinch assumed an elec-tron density, or tune shift, that linearly increases along thebunch. Simulations and analytical treatments show that inreality, due to the nonlinear oscillation of electrons in thebunch potential, “stripes” of high density form close to thecenter of the bunch and then propagate outwards [7]. Thepresence of a dipole magnetic field restricts the horizontalmotion of the electrons, and can lead to the appearance ofdifferent, “elliptical” stripes, that again start at the trans-verse bunch center and later shift outwards [8]. Recentstudies using a refined pinch model with stripes have un-covered a complex phase-space structure, indicating thepossibility of larger beam losses and stronger emittancegrowth than previously anticipated [6, 9].ELECTRON MOTIONIf the transverse beam size is much smaller than the vac-uum chamber, we can approximate the electron cloud den-sity in the vicinity of the beam prior to a bunch arrival by auniform distribution. Under the influence of the electricfield of the bunch, the electrons of the initially uniformcloud are perturbed and develop a structure with local den-sity enhancements.The electron motion in the bunch potential is character-ized by the linear oscillation frequencies of electrons closeto the transverse center of the beam, ωe;x,y . In the absenceof an external magnetic field and for a round bunch the fre-quency is the same in both planes, and, assuming a trans-verse Gaussian density with rms size σr, equal toωe(z) [m−1] ≡ λ(z)re/σ2r , (1)for an arbitrary longitudinal line density λ(z). Introducingthe radial coordinate r =√x2 + y2, and its normalizedcounterpart r˜ ≡ r/σr, the electron equation of motion isd2r˜dz2+ ω2e(z)r˜ = −ω2e(z)r˜(2(1− e− r˜22)− r˜2), (2)where the left-hand side represents the linear oscillationat small amplitudes, and the right-hand side the nonlin-ear terms. We observe that ωe(z) defines the scaling ofthe pinch, so that e.g., for a fixed longitudinal shape, dou-bling the bunch intensity is equivalent to halving the bunchlength, or to shrinking the transverse beam sizes by√2.If the bunch is not round, the horizontal and vertical os-cillation frequencies differ, and a second parameter or func-tion is needed to characterize the pinch, e.g. either twofrequencies ωe;x,y, or one together with the aspect ratioσy/σx.In case of a strong dipole field oriented in the vertical di-rection, we can consider the electrons’ horizontal positionas frozen. The vertical force yields the equation of motiond2y˜dz2+ ω2e(z)y˜ = −ω2e(z)y˜r˜2(2(1− e− r˜22)− r˜2), (3)where we have introduced a normalized vertical coordi-nate y˜ ≡ y/σr, and again assumed a round beam (σr ≡σx = σy). Also here ωe(z) characterizes the electron mo-tion completely, via (3).For comparing results it is convenient to introduce thelinear oscillation phaseφe(z) ≡∫ z−∞ωe(z′)dz′ . (4)which for two specific longitudinal profiles translates toφe(z) =reNb2σ2r{ (1 + erf(z√2σz))(Gaussian)(1 + 2z/lb) (uniform).SIMULATION PARAMETERSThe simulation employs about 100,000 macro-electronswhich are launched evenly distributed in the transversespace on a wide rectangular grid extending to ±30σx,ywith initial velocity. The initial electron energy of a fewelectron-volt can be neglected, since the typical electronkinetic energy acquired during the pinch is much larger, oforder me(cωeσr)2/2, where me denotes the electron massand c the speed of light. For simplicity, we will consideronly circular symmetric bunches, for which the electronmotion is described by (2) or (3). The bunch parametersin our simulation represent the LHC beam at injection: Weconsider Nb = 1.15× 1011 protons per bunch, with trans-verse rms size σx,y = 0.88 mm and rms bunch lengthσz = 11.4 cm. The transverse beam distribution is taken tobe Gaussian; in the longitudinal plane we choose either aGaussian or a uniform shape. The zero of the longitudinalcoordinate z coincides with the bunch center.STRIPE STRUCTUREFigure 1 presents the simulated density enhancement inthe x−z plane at y = 0 (left pictures) and also in a parallelplane with 2σ vertical offset (right pictures). The top pic-tures show results for a field-free region, the bottom onesfor a dipole field. In all cases, about 4 stripes emerge dur-ing the passage of the bunch. For the field-free region theelectron density at the center of the bunch, at y = 0, be-comes very high (note the different density scale). For aplane with vertical offset, y = 2σy, the stripe structure be-comes clearly visible also in the field-free case. Figure 2shows the corresponding density in the x − y plane at thelongitudinal position z = +σz .-3 -2 -1      0      1       2      3 -3 -2 -1      0      1       2      3 -3 -2 -1      0      1       2      3 -3 -2 -1      0      1       2      3 Figure 1: Electron density enhancement in the x− z planeat y = 0 (left) and y = 2σr (right) in a field-free region(top) and in a dipole (bottom), for a Gaussian bunch.Figure 2: Electron density enhancement in the transverseplane at z = σz without field (left) and in a dipole (right).Analyzing these data, the left picture of Fig. 3 presentsthe simulated vertical position of the outermost (horizontal)stripe as a function of its horizontal position. The stripe forthe dipole is almost of the same round circular shape inthe x − y plane as the one without field, and only slightlyflatter. The right picture shows that the density in the stripehardly varies with vertical position, both without field andin a dipole.dipoleno fielddipoleno fieldFigure 3: Vertical vs. horizontal position of the outermoststripe at z = 1σz , i.e. φe ≈ 1.45 × 2π (left), and the peakdensity in this stripe vs. the vertical position (right), com-paring a field-free region and a dipole magnet.Simulations were also performed for a longitudinal uni-form bunch profile with a full bunch length chosen equalto√2πσz . Figure 4 demonstrates that, when plotted as afunction of φe instead of z, the spatial distribution and mag-nitude of the density enhancement are similar, albeit notfully identical, for the Gaussian and uniform longitudinalprofiles. Always a new “stripe” emerges on axis roughlyat every half period of linear oscillation, starting from π/2,i.e. at φe = π/2, 3π/2, etc.Figure 4: Electron density enhancement in the x−φe plane,at y = 0, with a Gaussian (left) or uniform longitudinalprofile (right) without field (top) and in a dipole (bottom).DISCUSSIONComparing the top and bottom pictures in Figs. 1 or 4,we notice that while in a field-free region the electron den-sity of a stripe decreases as the latter “moves” to largeramplitudes, in a dipole field the density increases instead.This is further illustrated in the right picture of Fig. 5 whichshows the simulated density in the outermost stripe, in theplane y = 0, as a function of its horizontal position. Thetwo lines were fitted by eye through the simulation data foramplitudes xˆ > 1σr. They correspond to a linear peak den-sity evolution of ρˆ/ρ0 ≈ (17− xˆ/σr) in a field-free region,and ρˆ/ρ0 ≈ (11 + xˆ/σr) inside a dipole.The physical origin of the stripe patterns differs for thefield-free region and for the dipole field, as is illustrated inFig. 6. Without magnetic field, the electrons move radi-ally. At large amplitudes they undergo a highly nonlineardipoleno fieldtheoretical.linedipoleno fieldFigure 5: Horizontal position of the outermost stripevs. electron oscillation phase φe along the bunch (left), andthe density enhancement in this “stripe” as a function ofhorizontal stripe position (right), for a field-free region anda dipole. The solid line on the left is the prediction derivedfrom (7); the other three lines are “eyeball” fits.motion, losing their synchronization (left picture of Fig. 6).The stripes in physical space are the result of projecting theelectron distribution winding around the phase-space originonto the x axis. To estimate the amplitude of the turnoverpoint we may consider electrons which started their jour-ney at r˜0  1, so that we can approximate the force in (3)by its asymptotic form∝ 1/r˜. Integration yields∫ r˜r˜0dr′√ln (r˜0/r′)= −√πr˜0 erf(lnr˜0r˜)= 2φe(z) . (5)An inversion of this relation would give r˜ as a functionof r0, and the extreme point of this function r˜(r˜0) bean estimate for the location |x| of the outermost stripe.More empirically, the simulation data for an intermediaterange of φe values, 0.4<∼ φe <∼ 1, can be described by|x|/σr ≈ 3φe(z)/(2π), which is shown by a thin dashedline in the left picture of Fig. 5; for larger values of φe thestripe distance from the origin grows faster than linearly.After crossing the beam axis the electrons spread out uni-formly in all directions, and the peak density decreases in-versely with distance r˜, which may explain the shrinkingelectron density in the right picture of Fig. 5.x [mm]x’[1]outer ringinner rings-20 -10 20100-10050-501000outer ringinner ringsx [mm]y[mm]Figure 6: Snapshot at z = 1σz , of electrons in horizontalphase space for a field-free region (left) and in the x − yplane for a dipole field (right). The two colors on the leftdistinguish electrons which started on either side of thebeam. The right picture shows electrons launched at y > 0.In a dipole field, as the horizontal amplitude |x| increasesthe vertical electron motion becomes approximately linearover a larger and larger vertical range, |y| <∼ |x|, and, there-fore, an ever greater number of electrons reach the centralplane y = 0 simultaneously. The regions of increased den-sity on the horizontal axis correspond to the crossing ofy = 0 by large groups of electrons oscillating in the linearvertical beam field at x positions where the local verticalphase advance φe(x, z) is equal to π/2 plus a multiple of π(right picture of Fig. 6). More specifically, for amplitudes|y| <∼ |x| and |x|  σr, (3) can be approximated asd2y˜dz2+2ω2e(z)(x/σr)2y˜ ≈ 0 , (6)i.e., the local vertical oscillation frequency depends on x asωe,y(z, x) ≈√2ωe(z)σr/|x| , (7)where ωe(z) is the central oscillation frequency (1). Ac-cordingly, the location of a “stripe” starting at φe(z) ≈kπ/2 (k integer) near the origin, should later, atlarger horizontal amplitudes, be described by |x|/σr ≈4√2φe(z)/(2π)/k. The location of the outermost stripe(k = 1) is expected at |x|/σr ≈ 5.66φe(z)/(2π), which,in the left picture of Fig. 5, is superimposed on the dipolesimulation data as a thick solid line. Data and analyticalcurve are in nearly perfect agreement. The region wherethe vertical motion is approximately linear and, therefore,also the number of “synchronized” electrons crossing they = 0 plane at the same time grow in proportion to thedistance |x|, which may explain the density evolution for adipole field seen in the right picture of Fig. 5.CONCLUSIONSThe accumulated phase advance of the linear electron os-cillation φe(z) determines the spatial structure of the elec-tron pinch, almost independently of the longitudinal bunchprofile. The pinch structure is also affected by the presenceor absence of a magnetic field. In all cases considered, thehigh-density “stripes” are approximately circular rings inthe x − y plane. At every phase advance value φe equal toa π/2 pluS a multiple of π a new stripe emerges close tothe beam axis. The physical origin of the stripes is differ-ent in the field-free and dipole case, which explains why ina field-free region the peak electron density decreases as a“stripe” shifts outwards, while in a magnetic field the peakelectron density grows at larger amplitudes. For either casethe simulation indicates a linear dependence of the stripedensity on its distance from the axis, with a slope of +1 or−1, respectively, in normalized units. An analytical func-tion describes the variation of the horizontal stripe positionwith φe for a dipole field in good agreement with the simu-lation. The same dependence for the field-free case can bemodelled in general analytical terms and by an eyeball fit.REFERENCES[1] G. Arduini et al. Proc. PAC2001 Chicago, p. 685 (2001).[2] M.A. Furman, A.A. Zholents, Proc. PAC 99, p. 1794 (1999).[3] E. Benedetto et al, PRL 97:034801 (2006).[4] F. Zimmermann, Proc. Chamonix XI, p. 144, CERN-SL-2001-003-DI (2001); Proc. PAC 2001, p. 666 (2001).[5] K. Ohmi, Memorandum, November 2002; see http://ab-abp-rlc-ecloud.web.cern.ch/ab-abp-rlc-ecloud[6] G. Franchetti et al, Proc. CARE-HHH-APD BEAM’07,CERN, Oct. 2007.[7] E. Benedetto et al, Proc. ECLOUD’04, Napa, CERN-2005-001, p. 81l; Proc. EPAC’04, Lucerne, p. 1834 (2004).[8] E. Benedetto, PhD thesis, Politecnico Torino (2006).[9] G. Franchetti, “Incoherent Effects of Space Charge andElectron Cloud,” EPAC’08 Genoa (2008).

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